Solving the Schrödinger Equation with Numerov’s Algorithm

The Schrödinger equation describes the energy and time-evolution of a particle or system of particles, and is one of the fundamental building blocks of modern physics. In it’s general form, the (time-independent) Schrödinger equation looks like this:

\(\frac{-\hbar^2}{2m} \frac{\partial^2}{\partial x^2}\psi(x) + V(x) \psi(x) = E\psi(x)\)

There are relatively few situations in which the Schrödinger equation can be solved analytically, and numerical methods and approximations are one way around that analytical limitation. To demonstrate how this is possible and how a numerical solution works, what better way than to solve a system which can be solved analytically and comparing the results. Continue reading

Virtualizing Your Web Development Environment With VirtualBox

If you’re a part-time developer or you don’t use your computer exclusively for web development, it makes sense for you not slowing things down with Apache and MySql every time you boot up.

Aside from grinning and bearing it, one option to solve this would be to start these services when you need them from the command line. It’s a fairly simple sudo service apache2 start in most Linux distributions, or the new sudo start apache2 way for Ubuntu.

An alternative, which has its own advantages, is to install your services in a virtual machine and load up the VM when you’re working.

Actually, it’s perfectly possible to use this set-up for more than just single-person local development, and many developers do this.

One great advantage a virtualized development environment has is that it Continue reading